A Weighted Variant of Riemann-Liouville Fractional Integrals on ℝ n

Abstract

We introduce certain type of weighted variant of Riemann-Liouville fractional integral on ${ℝ}^n$ and obtain its sharp bounds on the central Morrey and $\lambda$-central BMO spaces. Moreover, we establish a sufficient and necessary condition of the weight functions so that commutators of weighted Hardy operators (with symbols in $\lambda$-central BMO space) are bounded on the central Morrey spaces. These results are further used to prove sharp estimates of some inequalities due to Weyl and Cesàro.